There are lots of classical pieces called "Sonata", and they usually have 3 movements. But there are guitar Sonatas by Paganini that only have 2 movements! And I heard there's a crazy Tchaikovsky -- maybe it was a symphony, but that's still sonata-form, right? -- piece that has 4 movements. I heard orchestras get spooked-out play this piece in Japan because they alone, having done their homework, know not to clap after the boisterous III. Allegro.

Would someone explain what the 'Sonata-Allegro' form is, and perhaps shed some light on these side-questions by-the-by?

[PS. Willing to separate this part as new question if appropriate.]

There's also the Rondo (== 'Rondeau'?) form that shows up in the same Piano collections. Are there others that belong to the same class?

1 Answer
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There are three different things here: the sonata(-allegro) form, the (multi-movement) sonata form, and the title sonata.

The sonata(-allegro) form is a form of one movement. It's usually fast (hence the allegro) and the big structure is ABA, where the first A is called exposition, B is development, and the second A is recapitulation. Sometimes there's a slow introduction and, especially with Beethoven and later composers, a coda at the end. In the most classic form, the exposition introduces a main theme, a secondary theme (in a different key), and a closing theme. In the development section the themes are varied and taken through different keys. The recapitulation is then a (varied) repetition of the exposition, often with the secondary theme transposed to the main key.

The (multi-movement) sonata form is a different thing, and refers to the structure of a whole composition. This (together with the sonata-allegro form) only really developed in the classical era. The classical multi-movement sonata usually consists of three or four movements:

A fast movement in sonata-allegro form.

A slow movement.

A menuetto or scherzo.

A fast finale, often in sonata-allegro or rondo form.

In three-movement sonatas the missing movement is usually one of the middle ones. Now, this structure is not only used for pieces called a sonata, but also for symphonies, concerti, trios, quartets, and other multi-movement "big" pieces.

Then there is the "Sonata" title. Originally, any instrumental piece could be called a sonata, for example the sonatas of Scarlatti. In the classical era, most pieces called a sonata were pieces of multi-movement sonata form written for one or two instruments. Since then we've been going back to the original meaning. I think if you want to describe a "sonata" in a way that encompasses most sonatas written after the classical era, you would have to say that it's a not-small piece for at most a few instruments. Usually it's also in more than one movement, and the form of the first movement quite often resembles the sonata-allegro form.

As you can see above, the rondo form can appear as a part of a multi-movement sonata. Rondo is usually of ABACADA or similar form, that is, there is a repeating main theme and then contrasting themes in-between. It may also be symmetric, for example ABACABA. The rondo form is not tied to the sonata form, though, and it often appears in independent pieces (the rondo form is actually much older than the sonata). Also, the finale of a classical multi-movement sonata can be something else than mentioned, for example a set of variations.